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The handle
http://hdl.handle.net/1887/83484
holds various files of this Leiden
University dissertation.
5 Time-resolved measurement of
electronic temperatures in a
single gold nanoparticle
5.1 Introduction
5.1.1 Background
Plasmonic nanoparticles are remarkable in their ability to couple to light whose wavelength is significantly larger than their size. This property is primarily due to their localized surface plasmon resonance — a resonance of collective electron oscillations enabled by the spatial confinement of the electron gas.
Leaving aside the rich physics created by various nanoparticle geometries, from tunable spectroscopic properties to powerful near-field effects, absorp-tion of light by a plasmonic nanosphere can be understood in the following simplified way:
If the dielectric permittivity of the material is known, the key parameters of scattering and absorption can be calculated using Mie theory. For far-field interactions, these are conveniently given in terms of cross sections: the scattering cross section 𝜎sca, the absorption cross section 𝜎abs, and the extinction cross section 𝜎ext = 𝜎sca+ 𝜎abs. Through 𝜎abs, we have access to the rate at which energy is absorbed by the nanoparticle, and thus to the amount of energy that must (eventually) leave the nanoparticle when it equilibrates with its environment.
The vast majority of the energy absorbed is converted to heat, which the (now hot) nanoparticle will dissipate. A vanishingly small fraction of the energy absorbed goes towards radiative emission channels referred to as photoluminescence; while this emission is measurable and useful — and indeed central to the technique used in this chapter — for the purposes of the energy balance, it can safely be neglected (𝑃pl ≪ 𝑃heat). This approached based on scalar cross-sections is the approach taken in the previous chapters of this thesis.
While this simple picture works well for low-electric-field near-steady-state continuous-wave excitation, it ignores the dynamics of light absorption, and the ultrafast heat transfer dynamics which occur in the metal. These come to the fore in particular when a plasmonic nanoparticle is excited with a pulsed laser with a pulse width below about a picosecond.
5.1 Introduction
as temperature fields varying in space. In nanoparticles that are sufficiently small compared to the wavelength and focus size of the heating laser, we can assume uniform heating. The temperatures of the electrons and of the lattice can then be taken to be constant across the entire volume of the particle, and may be treated as scalars.
In the two-temperature model, under ultrafast laser excitation, we typically assume the electrons are heated to some temperature 𝑇𝑒(𝑡 = 0) instantaneously. This requires that the temperature be well-defined, i.e. that the electrons thermalize to a Fermi–Dirac distribution with no significant delay. The charac-teristic thermalization time is of order ∼0.5 ps in gold [22, 111]; however, the electron distribution being instantaneously non-thermal barely has an impact on most measurements, especially if the excitation pulse width is of the same order. Assuming instantaneous thermalization, we model the evolution of the temperatures using the following coupled ordinary differential equations [105]:
𝐶𝑒(𝑇𝑒)d𝑇𝑒
d𝑡 = −𝐺(𝑇𝑒− 𝑇𝑙) (5.1)
𝐶𝑙d𝑇d𝑡𝑙 = 𝐺(𝑇𝑒− 𝑇𝑙) − 𝑄bath(𝑇𝑙), (5.2) where 𝑇𝑒and 𝑇𝑙are the temperatures of the electrons and the lattice, 𝐶𝑒and 𝐶𝑙are the corresponding heat capacities, 𝐺 is an electron–phonon coupling constant, and 𝑄bathrepresents heat transfer from the lattice to the environment. Note that the electronic heat capacity depends on the electronic temperature. The electronic excitation generally relaxes within a few picoseconds, while the lattice cools down over a longer period of time, which is limited by the thermal properties of the environment, especially the thermal conductivity and, for small particles in particular, the interfacial thermal resistance. Under typical conditions, relaxation of the lattice takes place over a period of hundreds of picoseconds.
The two-temperature model, by allocating the electrons to a Fermi distri-bution with a certain elevated temperature 𝑇𝑒 at time zero, does not take into account how the electrons got there. Considering this is important for two main reasons: Firstly, if the excitation laser pulse width is comparable to the electron–phonon coupling time, then they must, in principle, be con-sidered together. Secondly, there is no reason to believe that optical excitation of the electron gas yields a thermal distribution of electrons, and this must be considered if the excitation laser pulse approaches the electron–electron coupling time. Indeed, several studies have considered non-thermal electron distributions and how they thermalize [22, 114–116].
5.1.2 Anti-Stokes emission as a measure of temperature
When a nanoparticle is illuminated at a suitable wavelength, it can be detected optically, either through elastic (Rayleigh) scattering, in which the detected photons have the same energy as the illumination, or through inelastic pro-cesses, in which the detected photons have either gained energy from or lost energy to the nanoparticle. If the emitted photons have a higher energy (blue-shift), this is known as an anti-Stokes process, if they have a lower energy (red-shift), a Stokes process.
The simplest such inelastic process would be a single scattering event, in which a single incoming photon interacts once with a single particle (such as an electron, a hole, or a phonon) in the material, exchanges some energy and momentum with it, and is reemitted. Such lowest-order processes are collectively known as Raman scattering. A wide variety of higher-order processes may also occur: an interaction may involve two or more photons, or two or more electrons. The photon or photons may create an excited state, and this exciton may then decay radiatively some time later after some number of interactions with electrons, holes, and phonons.
5.1 Introduction
In metal nanoparticles specifically, excited states (electron-hole pairs, and, collectively, localized surface plasmon polaritons) are available, but they have relatively short lifetimes on the order of tens of femtoseconds [117]. This means that they do not dominate and that the excitons only experience a limited number of interactions with the thermal bath. In other words, the excited states decay radiatively before they have fully thermalized with their environment — this is known as ‘hot’ luminescence.
In general, certainly in metals, Raman scattering and hot luminescence will always occur together [118], and cannot easily be distinguished from one another in practice. In any case, energy is gained and lost through interactions with the thermal bath. Thus, anti-Stokes emission, in which the emitted photons have an energy ℏ(𝜔𝐿+ 𝛿𝜔), which is larger than the energy ℏ𝜔𝐿of the laser illumination, can only occur to the extent that occupied states with an energy of 𝜖 = ℏ𝛿𝜔 from which energy can be extracted are present in the metal. Both phononic and electronic states may, in principle, contribute, to these processes1.
The emission spectrum will depend on on the energy distribution of oc-cupied ‘hot’ states which might donate energy, 𝑛(𝑇 , 𝜖). Broadly speaking, this will follow a Boltzmann-type distribution: If hot electrons dominate the interactions, which is likely at large 𝛿𝜔 due to the higher temperature achieved by the electrons, it should obey Fermi–Dirac statistics. If, on the other hand, interactions with phonons dominate, it should follow Bose–Einstein statistics. In practice, the differences between the predictions of the three distributions are marginal for range of 𝛿𝜔 which we have access to experimentally2. The emission can then be used to deduce an electronic temperature which matches up well with a priori predictions [119, 120] and, by extrapolating to zero laser heating, to measure the temperature of the environment [121].
In the following, the term ‘photoluminescence’ will be used for inelastic emission in a broad sense, without prejudice as to the mechanism which causes it.
1The terminology of Raman scattering is a case in point: ‘traditional’ Raman scattering
exchanges energy with vibrational states, while electronic Raman scattering is universally couched in terms of scattering with quasi-free electrons, which dominate the behaviour of metals
2Nota bene, other authors have made the same observation about their measurements.
5.2 Method
5.2.1 Premise
We probe the dynamics which occur in response to pulsed irradiation of a gold nanosphere using time-resolved anti-Stokes spectroscopy; to achieve ultrafast time resolution, we use a two-colour pump-probe technique:
Two ∼ 350 fs laser pulses are sent to the sample with a particular delay 𝜏 between one and the other. The first, the ‘pump’ pulse, with a central wavelength of 𝜆 = 785 nm, is far to the red of the localized surface plasmon resonance of the nanosphere. The second, the ‘probe’ pulse, at 𝜆 = 594 nm, is near the resonance. We acquire emission spectra in the neighbourhood of 594 nm using spectral filters to remove the signal from the lasers and from elastic scattering.
Both colours are absorbed by the nanosphere, though the pump beam is absorbed significantly less efficiently. What’s crucial is that since the pump laser is ca. 0.5 eV to the red of both our observation range and the plasmon resonance, (i) any possible emission is not enhanced by the plasmon, and (ii) whatever inelastic emission may be detectable without plasmon enhancement is spectrally separated from the probe signal. In other words, the pump is, on its own, effectively invisible.
The pump pulse creates some excited population of hot electrons Δ𝑛ir(𝑡, 𝜖). The probe pulse, arriving a delay of 𝜏 later, also creates an excited population of hot electrons, Δ𝑛vis(𝑡 − 𝜏, 𝜖). However, its anti-Stokes emission depends on the total populations of electrons, which we may write as
𝑛(𝑡, 𝜖)||𝑡=𝜏= [𝑛0(𝜖) + Δ𝑛vis(𝑡 − 𝜏, 𝜖) + Δ𝑛ir(𝑡, 𝜖)]𝑡=𝜏 (5.3) if higher-order non-linear effects are small enough for the contributions of the two pulses to be additive. By varying the delay 𝜏 between the pump and probe pulses, as sketched in fig. 5.1, and examining the anti-Stokes spectra due to the probe, we can then elucidate the dynamics of Δ𝑛ir(𝑡, 𝜖) and the electron temperature with sub-picosecond time resolution.
5.2.2 Experimental details
5.2 Method
pump probe
time
𝜏 = −1 ps 𝜏 = −0.5 ps 𝜏 = +0.5 ps 𝜏 = +1 ps
Figure 5.1: Visual representation of different pulse delays 𝜏. Note that,
typic-ally, 𝜏 ≥ 0 for the pump pulse to have an effect on the measurement with the probe.
a central wavelength of 785 nm (the ‘pump’ pulses), and with a frequency-doubled optical parametric oscillator (OPO), which is pumped by the Ti:Sapph laser. The fact that one laser pumps the other means that every pulse from the OPO corresponds to a Ti:Sapph pulse — the two are locked together. The visible output of the OPO is tuned to 594 nm (the ‘probe’). Both pulses indi-vidually pass suitable dielectric band-pass filters which are well-matched to the notch filters in the detection. Additionally, we use a solid state continuous-wave 532 nm laser to identify particles, for fine adjustments, and for CW photoluminescence spectra.
The pulse width is measured to be equal to approximately 350 fs after an acousto-optic modulator in the beam path, which considerably lengthens the pulse. This component is needed only for the acquisition of transient extinction time traces (cf. chapter 4). Further optical components other than the objective should have little to no further effect on the pulse width; we can estimate the effect of the objective by approximating it as a solid block of glass with a length of ∼5 cm; for BK7 glass, this would result in a negligible lengthening of the pulse by 0.06 fs (at 785 nm) to 0.14 fs (at 594 nm). The equivalent values for other glasses are similar.
Figure 5.2: Highly simplified sketch of the experiment showing the logical beam paths.
being focussed on a fast photodiode (FEMTO Messtechnik). This arrangement allows measuring the change in extinction of the visible pulse as a function of inter-pulse delay and the acquisition of transient extinction time traces (as in chapter 4), but is further not required for the work discussed here. The pulse overlap is optimized using pump-probe extinction contrast (for the two pulses) and photothermal contrast (to overlap the pulses with the CW laser).
Meanwhile on the near side of the sample, the light (back-scattering and photoluminescence) collected by the NA = 1.4 objective is split off into the confocal detection path with a 50:50 beam splitter. After passing through a confocal pinhole (50 µm) and a series of notch filters for both colours, the emitted light is sent either to an avalanche photodiode, used for focussing, or to a liquid N2-cooled spectrograph (Acton Research SpectraPro-500i). Fig. 5.2 shows a rough logical sketch of the beam paths from source to detection, disregarding most optical components.
Photoluminescence spectra are acquired with an integration time of 240 s, with the exception of some spectra recorded at very low or very high intens-ity.3 All spectra shown here have been normalized by the integration time. Before the acquisition of each spectrum, the particle is brought into focus by maximizing photoluminescence with an automatic focussing routine. This is done to compensate for any slight drift in the system that may occur over the course of a long measurement. During the acquisition, the inter-pulse delay 𝜏 is kept constant.
Quoted pulse energies are measured as average powers in the back focal plane, before the objective, as indicated in fig. 5.2. They are not normalized by the transmission of the objective or by the absorptivity of the sample.
3The very long integration time is necessitated by the low intensity of the signal from a
5.3 Results
The sample consists of 100 nm gold nanospheres (Nanopartz Inc.) spin-coated on a glass cover slip (Menzel). Spheres were chosen as other geomet-ries are more liable to reshape under pulsed illumination. The nanoparticles are very dilute; in all cases the nearest neighbour of the nanoparticle being studied was more than ∼2 µm away. The nanoparticles on the glass are further immersed in a reservoir of ultrapure water.
5.3 Results
5.3.1 Dependence of the electronic temperature on intensity
In order to establish clearly which parts of the resulting anti-Stokes spectra are due to heating by the probe pulse — in the language of eq. (5.3), in order to get an idea of 𝑛0(𝜖) + Δ𝑛vis(𝜖) — we first measure Stokes and anti-Stokes spectra of the nanoparticle using the probe alone, at different excitation powers. The spectra are shown in fig. 5.3.
In general terms, if the anti-Stokes emission results from interactions with a Boltzmann-form thermal bath, we can say that the intensity should follow
𝐼AS(𝛿𝜔) ∝ 𝑓 (ℏ𝜔)𝑔(ℏ𝜔) exp (−ℏ𝛿𝜔𝑘
𝐵𝑇 ) , (5.4)
where 𝑔(ℏ𝜔) represents a density of states and 𝑓 (ℏ𝜔) represents a probability of emission. The use of a Boltzmann factor here disguises the distinction between the electron and phonon thermal baths, and is only justifiable if ℏ𝛿𝜔 is sufficiently large, i.e. far from the laser. In our measurements, the spectral filter blocking out the laser largely obscures the energy range in which this approximation breaks down.
10 1 100 Pulse Energy [pJ] 0.1 0.2 0.3 0.4 [eV] 10 2 100 102 PL59 4 [a rb .u .] 0.2 0.4 0.6 0.8 [eV] 10 2 100 102 PL59 4 [a rb .u .] 550 575 600 625 650 [nm] 0.0 0.5 1.0 PL53 2 [n or m
.] Particle IIParticle III
500 550 600 650 [nm] 0.0 2.5 5.0 7.5 10.0 PL59 4 [a rb .u .] P II, 0.58 pJP III, 0.54 pJ 0.0 0.5 1.0 1.5 2.0 Pulse energy [pJ] 0 500 1000 1500 Tvirt [K ] Particle II Particle III 0.0 0.5 1.0 1.5 2.0 Pulse energy [pJ] 0 50 100 A [arb. u.]
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.3: Anti-Stokes spectra of particles ‘II’ (a) and ‘III’ (b) excited by
594 nm pulses; different colours represent different pulse energies (colour bar above). Dashed lines are fits to the Boltzmann distribution, eq. (5.5). (c) Photoluminescence spectra of the same particles excited at 532 nm (CW).
(d) Examples of spectra including the Stokes and anti-Stokes components,
5.3 Results
We resort instead to a rather simpler approximation: we assume that the function 𝑓 (ℏ𝜔)𝑔(ℏ𝜔) varies much more slowly than the exponential Boltzmann factor. This leaves us with
𝐼AS(𝛿𝜔) = 𝐴 exp (−ℏ𝛿𝜔𝑘
𝐵𝑇 ) , (5.5)
where 𝐴 is a proportionality coefficient.
Due to the lack of a sharp resonance, this approximation is reasonable sufficiently far from the interband transitions. The nearby interband transition at 470 nm [123] corresponds to ℏ𝛿𝜔 ≈ 0.5 eV (i.e. 6 × 103K × 𝑘𝐵), at which point we do indeed clearly see the fits to eq. (5.5) break down in fig. 5.3b.
The same approximation has been used successfully by other authors in the past, such as He et al. [124] and Cai et al. [119]. Authors inclined to interpret similar measurements as electronic Raman scattering, such as Crampton et al. [120], instead assume that 𝑓 (ℏ𝜔)𝑔(ℏ𝜔) ∝ 𝜔3, which is practically equivalent to our approximation since 𝜔3varies much more slowly than the Boltzmann factor.
This simplistic Boltzmann description fits the data well up to ℏ𝛿𝜔 ≈ 0.4 eV, at which point the approximations start to break down as expected. Note, however, that the extracted temperatures reflect the slope of logarithm of the data at lower values of 𝛿𝜔 than that, and are thus not particularly affected by the interband transitions.
The temperatures 𝑇 and amplitudes 𝐴, shown in fig. 5.3e and 5.3f respectively, broadly show the expected features:
(i) The temperatures reach >1000 K, and increase monotonically with in-creased heating power. The high temperatures indicate that we are indeed measuring the temperatures of the electrons, and not of the gold lattice.
(ii) The intensity of the anti-Stokes emission also increases with heating power, both due to the increased number of photons at higher powers, and due to the larger anti-Stokes to Stokes ratio at elevated temperatures. The combination of the two leads to a faster-than-linear increase.
shown theoretically to lead to a similar slowing of the change in temperature [125, 126].
(iv) The temperature roughly approaches room temperature as 𝑃 → 0. This is, however, far from exact for two main reasons. Firstly, extrapolating to zero power is not possible without greater knowledge of the nonlinear function 𝑇 (𝑃), and secondly, the margins of error are substantial: on the one hand, the rather low signal-to-noise ratio at the lowest powers increases the uncertainty as the temperature decreases, and on the other, the approximations inherent in eq. (5.5) limit the reliability of exact quantitative conclusions in the first place.
Evidently, a more detailed model of the emission mechanism as well as the thermodynamic and photothermal effects in this system is needed to accurately calculate and understand the true electronic temperature, and its behaviour as a function of power.
Fig. 5.3 shows a remarkable heterogeneity between the two particles: particle III appears to be heated significantly more efficiently than is particle II, even though the particles have nearly the same size; this implicates other factors that may influence the absorption at 594 nm and elsewhere, such as the precise shape of the particles (including surface facets), interactions with the substrate, or crystal defects.
5.3.2 Hot electron dynamics
In the two-colour experiment, spectra are recorded while focussing two pulse trains on the particles, a probe pulse which causes the measured emission, and a pump pulse, whose direct effect is invisible. The delay between the pulses 𝜏 is varied from one spectrum acquisition to the next. 𝜏 is calibrated such that at positive 𝜏 the probe pulse arrives after the pump pulse, and such that 𝜏 = 0 corresponds to the initial maximum of a ‘traditional’ pump-probe extinction time trace (such as those in chapter 4).
Fig. 5.4 shows such time-resolved spectra for three different 100 nm (nom-inal) gold nanospheres. The effect of the presence of the pump pulse on the spectra is limited to a period of ∼1 ps; at longer delays the emission returns to its initial state. The change on the anti-Stokes side of the spectrum is significantly greater than the change on the Stokes side.
5.3 Results
(a) – particle I (b) – particle I
550 600 [nm] 0.0 2.5 5.0 7.5 [p s] 0.0 0.1 0.2 0.3 PL59 4 [a rb .u .] 550 600 [nm] 0.0 2.5 5.0 7.5 [p s] 0.1 0.0 0.1 PL59 4 [a rb .u .]
(c) – particle III (d) – particle III
550 600 [nm] 2 0 2 4 [p s] 0.0 0.5 1.0 1.5 PL59 4 [a rb .u .] 550 600 [nm] 2 0 2 4 [p s] 0.50 0.25 0.00 0.25 0.50 PL59 4 [a rb .u .]
(e) – particle V (f) – particle V
550 600 [nm] 2 0 2 4 [p s] 0 1 2 3 PL59 4 [a rb .u .] 550 600 [nm] 2 0 2 4 [p s] 1 0 1 PL59 4 [a rb .u .]
Figure 5.4: Time-resolved spectra of three different 100 nm (nominal) gold
emission due to elevated electron temperatures are due only to secondary effects, such as the change in the dielectric constant.
As stated in § 5.2.1, the measured anti-Stokes spectra are characteristic of the total electron distribution at the moment the probe pulse arrives, as given in eq. (5.3): they are characteristic of 𝑛(𝑡, 𝑇 , 𝜖)|𝑡=𝜏. Here, two approaches for analysis present themselves:
For a straightforward parametrization, we can simply fit eq. (5.5) to the spectra for each time 𝜏. The equation, now
𝐼AS(𝜏, 𝛿𝜔) = 𝐴(𝜏) exp (𝑘−ℏ𝛿𝜔
𝐵𝑇 (𝜏)) , (5.6)
immediately gives us two time-dependent parameters that characterize the resulting emission quite well: a temperature 𝑇 (𝜏) and a quasi-amplitude 𝐴(𝜏).
Alternatively, we can make use of the fact that [𝑛0(𝜖) + Δ𝑛vis(𝑡 − 𝜏, 𝜖)]𝑡=𝜏, i.e. the effect of the probe pulse alone, is independent of the inter-pulse delay 𝜏 to our advantage. Figs. 5.4b, 5.4d and 5.4f show the difference spectra
Δ𝐼AS(𝜏, 𝛿𝜔) = 𝐼AS(𝜏, 𝛿𝜔) − ̂𝐼AS(𝛿𝜔) (5.7) if we subtract this 𝜏-independent component, the baseline spectrum ̂𝐼AS(𝛿𝜔). For the purpose of parametrizing the entirety of the data, this approach is less practical since Δ𝐼AS(𝜏, 𝛿𝜔) is overwhelmed by noise after a picosecond or two. The parameters 𝐴(𝜏) and 𝑇 (𝜏) resulting from a fit to eq. (5.6) are shown in fig. 5.5, along with the corresponding pump-probe extinction spectra from which 𝜏 = 0 is calibrated. The magnitude of the change in the anti-Stokes spectra, represented by 𝐴(𝜏) follows the behaviour of the change in extinction well: Both have their maxima at the same inter-pulse delay, and both show the same asymmetric behaviour as a function of 𝜏, with a steep rising edge as the pulse is absorbed, and a slower ∼ ps decay as the absorbed energy is released into the metal lattice.
The highest apparent temperature state is, however, reached earlier. Across our measurements, temperature peaked around (0.66 ± 0.10) ps before the amp-litude, which is almost twice our pulse width of 350 fs. The high-temperature state then rapidly decays to its initial value as the anti-Stokes intensity in-creases.
5.3 Results
(a) – particle I (b) – particle III
2.5 0.0 2.5 5.0 7.5 [ps] 0 1 ext. / max 600800 1000 1200 T [K] 0.2 0.4 0.6 A [arb.u.] 2 0 2 4 [ps] 0 1 ext. / max 800 1000 1200 T [K] 1 2 A [arb.u.] (c) – particle V (d) 2 0 2 4 [ps] 0 1 ext. / max 700 800 900 1000 T [K] 2 3 4 A [arb.u.] 550 575 600 625 650 [nm] Particle III Particle I Particle V
Figure 5.5: (a–c) Parametrization of the spectra in fig. 5.4 in terms of 𝐴(𝜏)
500 525 550 575 600 625 [nm] 0.0 0.2 0.4 0.6 luminescence change -1.8 ps -0.6 ps 0.0 ps +0.6 ps +2.4 ps
Figure 5.6: Difference spectra from fig. 5.4f: In order of increasing 𝜏, the state
before the pump pulse, the highest 𝑇 state (𝜏 = −0.6 ps), the highest intensity state (𝜏 = 0), a slightly later state (𝜏 = 0.6 ps) and a late-𝜏 state.
5.4f for selected delays 𝜏. The highest-𝑇 spectrum, 𝜏 = −0.6 ps, clearly decays much more slowly with decreasing 𝜆 than does either the tallest spectrum at 𝜏 = 0 or indeed any later spectrum, such as the example from 𝜏 = +0.6 ps.
The fact that the state of maximum temperature occurs so much earlier than the state of peak response in terms of both extinction and luminescence calls into question, at the very least, the implicit assumption that the peak of the pump-probe extinction spectrum, which we refer to as 𝜏 = 0, corresponds to the pump and probe pulses arriving at the same moment.
The temperature changes due to the pump pulse as reported in fig. 5.5, some 300 K or so, appear quite low. This is an artefact caused by the under-lying assumption of eq. (5.6): for the fit to give a good understanding of the temperatures, the entire hot electron gas would have to have a well-defined temperature, i.e., would have to be fully thermalized. In terms of eq. (5.3), it would require Δ𝑛irto have equilibrated with Δ𝑛visand 𝑛0before the emission of any photons.
As suggested above, we can get closer to that picture of the electron dis-tribution by subtracting a baseline spectrum ̂𝐼AS(𝛿𝜔) from the full spectrum 𝐼AS(𝜏, 𝛿𝜔) to arrive at the effect of the pump pulse alone, without the effect of the probe pulse. Just as with the full spectrum, we may then imagine it to be caused by a thermal distribution of electrons4and fit the spectra to a
4While we expect the initial distribution of hot electrons to be non-thermal [22, 114–116],
5.4 Discussion and conclusion
𝑇 from 𝐼AS 𝑇Δfrom Δ𝐼AS
Particle max 𝜏 = 0 max 𝜏 = 0
I 1278 K 1067 K 1661 K 1206 K III 1279 K 958 K 1756 K 1004 K
V 1015 K 868 K 1306 K 936 K
Table 5.1: Apparent temperatures of the total spectra (figs. 5.4a, 5.4c, 5.4e
and eq. (5.6)) and difference spectra (figs. 5.4b, 5.4d, 5.4f and eq. (5.8)) at maximum temperature and at 𝜏 = 0. The values for the former correspond to those in fig. 5.5. Note the spectra for particles III and V were measured on the same day and with the same pump pulse energy, viz. ca. 2.2 pJ as measured in the back focal plane.
Boltzmann distribution:
Δ𝐼AS(𝜏, 𝛿𝜔) = 𝐼AS(𝜏, 𝛿𝜔) − ̂𝐼AS(𝛿𝜔)= 𝐴! Δ(𝜏) exp ( −ℏ𝛿𝜔
𝑘𝐵𝑇Δ(𝜏)) . (5.8) The results of this fit for the high-𝑇 and high-𝐴 states are listed in table 5.1 alongside the corresponding values for the fit of the full spectra from fig. 5.5. It shows that the distributions created by the pump pulse initially have significantly higher characteristic temperatures than those present due to the probe pulse, which decays rapidly, as seen in the previous figures.
5.4 Discussion and conclusion
We have measured anti-Stokes photoluminescence of single gold nanoparticles due to pulsed illumination with characteristic temperatures of order 103K. The temperatures are extracted using a simple Boltzmann approximation (eq. (5.5)) that fits well for intermediate anti-Stokes shifts that are not so small that the Boltzmann distribution would cease to apply, and that are not so large that the interband transitions of gold become significant.
For a more exact and more precise extraction of the temperature from the photoluminescence spectra a more thorough model of the origin of gold nanoparticle photoluminescence is needed. Such a model may also provide more insight into the nature of the apparent temperature (i.e., is it entirely due to the electron temperature?). However, such a ‘full’ model should only amount to a modest correction to the deduced temperature.
In eq. (5.3) we assumed that the excited electron populations created by the two laser pulses are independent of one another. However, in fig. 5.3e, we see that dependence of the anti-Stokes spectra on pulse energy deviates significantly from linearity. This is not, in and of itself, terribly surprising: we know that the heat capacity of the electron gas and the dielectric permittivity both depend on temperature. The probe pulse energies used in the two-colour measurements are on the low side (∼0.1 pJ, see fig. 5.4 caption), so the linear approximation inherent in eq. (5.3) may still be reasonable, but we cannot exclude interactions between the pulses that would affect the interpretation of these measurements.
It may be interesting to extend fig. 5.3e to higher pulse energies to establish whether the temperature saturates under strong illumination before melting of the particle becomes an issue.
We have shown in § 5.3.2 that the apparent temperature increase due to pulsed excitation decays on a ∼ps timescale, which agrees with previous measurements of the electron-phonon coupling time. Surprisingly, the peak apparent temperature is reached early in the process (fig. 5.6), 0.6 ps before the peak amplitude. This time consistent with the thermalization times previously measured in bulk gold using time-resolved photoemission spectroscopy [111]. We note that what appears to be an early high-temperature state may in fact be a non-thermal state with a ‘hot electron’ contribution (which we measure) and a lower-temperature contribution (which is mostly obscured by the spectral filter near the laser).
This surprising observation may allow some deeper insight into the thermal-ization dynamics of the electron gas in a gold nanoparticle during the arrival of short laser pulses, if it were examined using a detailed model of the electron distribution and its evolution as a function of time.
5.4 Discussion and conclusion
plenty of time during the probe pulse for the electron distribution to approach a thermal one. Secondly, the spectral filter removing the laser obscures small energy shifts, which means we can’t compare the distributions at low and at high energies as extensively as we might like.
Either one of these shortcomings could be improved somewhat, but there is a trade-off between the two: If one wanted to work with a shorter pulse, this pulse would inevitably have a larger bandwidth (at least 0.2 eV for ∼10 fs, for instance), which would obscure more of the anti-Stokes spectrum. Vice versa, a much narrower filter would soon require the use of longer laser pulses.
